The paper is devoted to the construction of an analytical solution of the initial problem for the one-dimensional system of equations of non-Newtonian hemodynamics. In the model, the blood is considered as the power-law fluid. The perturbation method for the solution is proposed. The cases of the first and second perturbations are considered. The obtained solutions can be used for the comparison of different rheological models of blood or for the testing of programs, which realize the numerical algorithms.